Krylov-space anatomy and spread complexity of a disordered quantum spin chain

Bikram Pain; David E. Logan; Sthitadhi Roy

REVIEW 2 major objections 4 minor 2 cited by

Krylov-space anatomy and spread complexity of a disordered quantum spin chain

T0 review · 2 major / 4 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Long-time Krylov spread complexity scales linearly with Fock-space size in the ergodic phase and only sublinearly in the MBL phase, so the late-time state fills a finite versus vanishing fraction of the Krylov chain.

desk verdict Clean ED demonstration that infinite-time Krylov spread complexity scales as N_H vs N_H^\alpha (\alpha<1) and that MBL profiles are stretched-exponential, with a solid large-deviation picture of rare resonant eigenstates. read the letter →

arxiv 2603.25724 v2 pith:GIEJFGBB submitted 2026-03-26 cond-mat.dis-nn cond-mat.stat-mechquant-ph

Bikram Pain , David E. Logan , Sthitadhi Roy This is my paper · ORCID

classification cond-mat.dis-nncond-mat.stat-mechquant-ph

keywords Krylovspreadcomplexitymany-bodylocalisationergodicphasedisorderedspinchainstretchedexponentiallarge-deviationanalysisFock-spacedimensioneigenstateresonances

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how a many-body quantum state spreads when it is written in the ordered Krylov basis generated by its own Hamiltonian, and whether that basis-optimised spread cleanly separates the ergodic phase from the many-body localised (MBL) phase of a disordered spin chain. It finds that the infinite-time Krylov spread complexity grows linearly with the dimension of Hilbert space in the ergodic regime, so the late-time state occupies a finite fraction of the Krylov chain, while the same quantity grows only as a sublinear power of the dimension in the MBL regime, so the state remains confined to a vanishing fraction of the chain. The average probability profile along the chain itself decays as a stretched exponential in the MBL phase; a simple phenomenological model attributes the stretch to a broad distribution of exponential decay lengths across eigenstates. A large-deviation analysis of the individual eigenstate contributions confirms that almost every eigenstate participates in the ergodic sum, whereas only a vanishing (yet still exponentially large) fraction of rare, anomalously complex eigenstates dominate the MBL sum. The result supplies a transparent, one-dimensional geometric picture of how localisation and rare resonances appear once distance is measured in the Krylov basis rather than on the high-dimensional Fock-space graph.

What carries the argument

Krylov spread complexity S_{K,∞} = Σ_n n Λ_n, where Λ_n is the infinite-time probability of finding the state on the n-th Krylov orbital; this quantity is the first moment of a probability distribution on a one-dimensional chain of length equal to the Fock-space dimension and is basis-optimised by construction of the Krylov basis.

What would settle it

Exact diagonalisation of the same model at larger system sizes that extracts both the scaling exponent α of S_{K,∞} and the stretch exponent γ of ⟨Λ_n⟩; if α remains 1 deep in the putative MBL regime or if γ fails to approach the predicted asymptotic value 1/3, the central geometric claim is falsified.

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Extended reading notes

Core claim

Infinite-time Krylov spread complexity scales as N_H in the ergodic phase of the disordered tilted-field Ising chain and as N_H^α with α<1 in the MBL phase; the associated disorder-averaged profile Λ_n on the Krylov chain is flat (after a short transient) when ergodic and stretched-exponential when many-body localised, the latter arising because the infinite-time state is a weighted sum of exponentially decaying eigenstate amplitudes whose characteristic lengths are themselves broadly distributed.

Load-bearing premise

The analytic explanation for the stretched-exponential profile assumes that each eigenstate decays purely exponentially on the Krylov chain with a length drawn from a simple exponential distribution, and that those lengths themselves are exponentially distributed across disorder realisations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The punchline is simple: infinite-time Krylov spread complexity S_{K,\infty} scales linearly with Fock-space dimension in the ergodic phase and sublinearly (N_H^\alpha, \alpha<1) in the MBL phase of the tilted-field Ising chain, while the long-time profile \Lambda_n on the Krylov chain is stretched-exponential in MBL and essentially flat in the ergodic regime. The large-deviation entropy density \Sigma(x) of eigenstate complexities then shows that the MBL sum is dominated by a vanishing fraction of anomalously complex eigenstates.

What is new is the systematic finite-size scaling of S_{K,\infty}, the collapse of \langle\Lambda_n\rangle, the stretch-exponent analysis, and the multifractal IPR of the eigenstate contributions (Figs. 4–10, Table I). Earlier Krylov-complexity papers already used the measure as an ergodicity diagnostic; this work supplies the concrete geometric picture on the ordered Krylov chain and the rare-resonance statistics that make the distinction sharp. The ED data (L up to ~14–16, disorder averages, error bands) are internally consistent, the Gaussian-to-exponential change in the distribution of S_{K,\infty} is clean, and the connection to the authors’ earlier Fock-space work is properly cited rather than recycled as a new claim.

The soft spot is exactly the one the reader flagged: the phenomenological exponential distributions of decay lengths in Sec. IV C are chosen for analytic convenience and only a posteriori match the observed stretch exponents (\gamma~1/2 at accessible sizes, asymptotically 1/3). They are not load-bearing; the central scalings come straight from the Lanczos-generated Krylov basis of the microscopic Hamiltonian. Finite-size caveats on MBL numerics remain, but they are generic and already acknowledged. No code is shipped, yet the model and methods are standard and re-implementable.

This is for people who already work on MBL diagnostics or Krylov complexity and want a transparent geometric probe. It does not settle the existence of the MBL phase, but it is a solid, well-written extension that a serious referee should see. I would accept it for peer review and would cite the scaling and large-deviation results.

Referee Report

2 major / 4 minor

Summary. The paper studies the Krylov-space anatomy of states and the infinite-time Krylov spread complexity S_{K,∞} for the disordered tilted-field Ising chain, contrasting the ergodic and MBL regimes. Using the Lanczos-generated Krylov basis from a mid-spectrum product state, the authors show that ⟨S_{K,∞}⟩ scales linearly with Fock-space dimension N_H in the ergodic phase (occupying a finite fraction of the Krylov chain) and sublinearly as N_H^α with α<1 in the MBL phase (occupying a vanishing fraction). The disorder-averaged profile ⟨Λ_n⟩ collapses onto a scaling form with the same α and, in the MBL regime, exhibits a stretched-exponential decay. A large-deviation analysis of the eigenstate contributions S_{K,|E⟩} further shows that the ergodic sum is carried by a finite fraction of eigenstates while the MBL sum is dominated by a vanishing (but still exponentially large) fraction of anomalously complex eigenstates, with multifractal IPRs of those contributions. A phenomenological theory based on exponential length-scale distributions is offered to rationalize the stretch exponent.

Significance. If the reported scalings hold, the work supplies a clean, basis-optimized diagnostic that sharply separates ergodic and MBL phases on a one-dimensional Krylov chain whose length is exactly N_H. The combination of direct ED scaling of S_{K,∞} and Λ_n (Figs. 4–7), the large-deviation entropy density Σ(x) (Fig. 9, Table I), and the multifractal IPR of eigenstate complexities (Fig. 10) is internally consistent and goes beyond earlier operator-Krylov studies that are less sensitive to MBL. The explicit mapping between Krylov orbitals and Fock-space Hamming shells (Sec. III B, Fig. 2) also clarifies why the Krylov chain remains a faithful probe even when most orbitals have support over the entire Fock graph. Finite-size caveats on MBL numerics are generic and already acknowledged; the central numerical distinction itself is robust within accessible sizes.

major comments (2)

Sec. IV C, Eqs. (33)–(47): the phenomenological theory assumes pure exponential decay of contributing eigenstate amplitudes with lengths ξ_|E⟩ drawn from an exponential distribution of mean ξ_d, and that the ξ_d themselves are exponentially distributed over disorder. This functional form is chosen for analytic convenience and is only a posteriori consistent with the observed stretch exponents (γ ≃ 1/2 at accessible sizes, asymptotically 1/3). Because the central claims of the paper rest on the ED scalings of S_{K,∞} and Λ_n (Figs. 4–7) and on the large-deviation analysis (Sec. V), not on this ansatz, the theory should be clearly labeled as a post-hoc rationalization rather than a derivation. A short numerical check of the actual distribution of effective decay lengths extracted from individual eigenstates would strengthen or falsify the assumption.
Sec. IV A and Fig. 4: the reported exponents α(W) are extracted from system sizes L ≲ 14–16. While the ergodic α = 1 result is solid, the MBL values α < 1 (and the associated claim that the long-time state occupies a vanishing fraction of the Krylov chain) remain subject to the usual finite-size caveats of MBL numerics. The manuscript should state more explicitly the largest L used for each W, the number of disorder realizations, and whether any drift of α with L is visible; a brief comparison with an independent localization diagnostic (e.g., half-chain entanglement or Fock-space IPR) on the same samples would help calibrate how deep into the putative MBL regime the data sit.

minor comments (4)

Fig. 7 and surrounding text: the stretch exponent is quoted as γ ≃ 1/2 for the bulk of the data and γ = 1/3 for the largest n/N_H^α. A single sentence clarifying that the asymptotic analytic result is γ = 1/3 while finite-size data remain closer to 1/2 would remove residual ambiguity.
Appendix A, Fig. 11: the scaling of ⟨b_n⟩ and the effective disorder W_n are shown but not used later. Either a brief remark on why these bare Krylov-matrix statistics do not distinguish the phases, or a pointer to future work, would improve cohesion.
Eq. (9) and the definition of Λ_n: the sum rule ∑_n Λ_n = 1 is stated, but it would help the reader to note explicitly that the infinite-time average eliminates the off-diagonal E eq E' terms, so that Λ_n is strictly a sum of |c_n,E|^2 |c_0,E|^2.
References: a few recent works on state Krylov complexity near the MBL transition (e.g., those already cited as [54–59]) could be more explicitly contrasted in the introduction to highlight what is new in the infinite-time anatomy and large-deviation analysis.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: central scalings and large-deviation results are direct ED observables; the exponential length-scale ansatz of Sec. IV C is a post-hoc phenomenological rationalization, not a load-bearing derivation.

other [Sec. IV C, Eqs. (33)–(34), (43)–(44)]
"The two key ingredients that enter the phenomenological picture are: (i) For a given disorder realisation, there is a normalised distribution over eigenstates, P_ u|E( u), of lengthscales u_|E on the Krylov chain, of form P_ u|E( u)=1/ u_d imes p_ u|E( u/ u_d). u_d itself has a distribution P_ u d( u_d)=1/ u imes p_ u d( u_d/ u). u Motivated by this, we assume also the simplest such exponential distribution for P_ u|E( u), P_ u|E( u)=1/ u_d exp[- u/ u_d]."

The exponential forms are chosen by hand for analytic convenience after the stretched-exponential profile has already been observed numerically; they reproduce u=1/2 (or 1/3 after disorder average) by construction of the integral representation, but are not used to generate or force the primary ED scalings of S_{K, u} or u_n. This is a mild post-hoc rationalization rather than a circular derivation of the central claims.

full rationale

The paper’s primary claims (linear vs sublinear scaling of S_{K,\infty} with N_H, stretched-exponential profile of u_n, and large-deviation dominance by rare eigenstates) are extracted from exact diagonalization of the microscopic tilted-field Ising Hamiltonian after Lanczos construction of the Krylov basis (Secs. IV A–B, V; Figs. 3–10, Table I). These quantities are defined independently via Eqs. (6)–(9) and (48)–(49) and measured without intermediate fitting that is later re-labeled as prediction. The phenomenological theory of Sec. IV C posits exponential distributions P_ u|E and P_ u d solely to rationalize the observed stretch exponent u o 1/2 (finite-size) or 1/3 (asymptotic); the ansatz is not used to define or force the measured S_{K, u} or u_n, nor is it claimed to be first-principles. Self-citations to the authors’ prior Fock-space work supply background context and are not invoked as uniqueness theorems or load-bearing premises for the Krylov results. No self-definitional loop, fitted-input-as-prediction, or renaming of a known result appears. The single minor note is the a-posteriori choice of exponential forms, which does not elevate the score above 1.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The work rests on the standard definition of Krylov complexity (minimizing the cost function with µ_n=n), the existence of an MBL regime in the TFI chain for W≳3.7 at accessible sizes, and two phenomenological distributional assumptions introduced to explain the stretch exponent. No new particles or forces are postulated; free parameters are the usual disorder strength, system size, and the fitted exponents α, γ.

free parameters (3)

α (scaling exponent of ⟨S_{K,∞}⟩) = 1 (ergodic); <1, W-dependent (MBL)
Extracted from power-law fits of disorder-averaged infinite-time complexity versus N_H for each W; values decrease from 1 (ergodic) to ~0.57 (W=10).
γ (stretch exponent of ⟨Λ_n⟩) = ≃1/2 (finite size); 1/3 (asymptotic claim)
Read off from log-log plots of -ln f(x); consistent with 1/2 over accessible range, asymptotically argued to be 1/3.
critical disorder W_c = ≃3.7
Taken from prior literature as ≃3.7 for the chosen parameters; used only to label phases, not fitted here.

assumptions (3)

domain assumption The Krylov basis generated by H from |ψ_0⟩ minimizes the cost function C_V = Σ n |⟨ψ_t|V_n⟩|^2 among all ordered orthonormal bases.
Invoked in Sec. II via citation to Balasubramanian et al. (2022); taken as given rather than re-derived.
ad hoc to paper For each disorder realization the contributing eigenstate amplitudes on the Krylov chain decay exponentially with lengths ξ_|E| drawn from an exponential distribution of mean ξ_d, and the ξ_d themselves are exponentially distributed over disorder.
Introduced in Sec. IV C to obtain the stretched-exponential profile analytically; not derived from the microscopic Hamiltonian.
domain assumption The disordered tilted-field Ising chain hosts a many-body localized phase for sufficiently strong disorder at the system sizes studied.
Standard working assumption of the MBL literature; critical W≃3.7 quoted from Abanin et al. (2021).

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Pith. "Pith review of Krylov-space anatomy and spread complexity of a disordered quantum spin chain." pith.science (2026). https://pith.science/paper/GIEJFGBB

@misc{pith2026260325724,
  author       = {Pith},
  title        = {Pith review of: Krylov-space anatomy and spread complexity of a disordered quantum spin chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIEJFGBB}},
  note         = {Machine review of arXiv:2603.25724}
}

read the original abstract

We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic regime, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL regime, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL regime. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.

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Pith tools

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